# Expert Answers - The mean of the sampling distribution.

Questions 1 to 16 are multiple-choice questions. Select one (1) answer from the given choices. One mark will be awarded for each correct answer.

1. The mean of the sampling distribution of the sample mean is

(a) the mean of the means of all possible samples of the same size taken from the population

(b) the mean of the frequency distribution of the population

(c) the mean of the means of all frequency distributions

(d) the mean of one sample

2. To apply the central limit theorem to the sampling distribution of the sample mean, the sample is considered to be large if

(a) n is greater than 50

(b) n is 50 or larger

(c) n is larger than 40

(d) n is 30 or larger

3. The weights of all babies born at a hospital have a mean of 8.4 pounds and a standard deviation of 0.70 pounds. The mean of the sampling distribution of the mean weight of a sample of 49 babies born at this hospital is

(a) 1.2 pounds

(b) 8.4 pounds

(c) 0.10 pounds

(d) 7.0 pounds

4. The number of elements in a sample with a specific characteristic divided by the total number of elements in the sample is called

(a) the sample mean

(b) the sample proportion

(c) the sample distribution

(d) the sampling distribution

5. The value(s) assigned to a population parameter based on the value of a sample statistic is called

(a) the probability

(b) a probability distribution

(c) a sampling distribution

(d) an estimate

6. The Z value for a 95% confidence interval for the population mean is

(a) 2.07

(b) 1.96

(c) 2.17

(d) 1.65

7. A random sample of 100 customers who visited a department store spent an average of $77 at this store with a standard deviation of $19. The 97% confidence interval for the population mean is

(a) 74.56 to 79.44

(b) 70.18 to 83.82

(c) 73.89 to 80.11

(d) 72.88 to 81.12

8. A sample of 200 elements produced a sample proportion of 0.47. The applicable

Z score used in calculating an 98% confidence interval for the population

proportion is

(a) 2.323

(b) 0.765

(c) 1.388

(d) 2.862

9. In a test of hypothesis, the Type II error occurs when

a. a false null hypothesis is rejected

b. a true null hypothesis is not rejected

c. a false null hypothesis is not rejected

d. a true null hypothesis is rejected

10. A two-tailed test of hypothesis contains

(a) one rejection region and two nonrejection regions

(b) two rejection regions and one nonrejection region

(c) two rejection regions and two nonrejection regions

(d) one rejection region and one nonrejection regions

11. In a left-tailed test of hypothesis, the sign in the alternative hypothesis is

(a) not equal to ()

(b) greater than ()

(c) less than (<)

(d) less than or equal to ()

12. In a test of hypothesis, the null hypothesis is that the population mean is equal to 80 and the alternative hypothesis is that the population mean is less than 80. A

sample of 100 elements selected from this population produced a mean of 74 and

a standard deviation of 12. The value of the test statistic is

a. Z = 4.78

b. Z = 5.00

c. Z = -5.00

d. Z = -4.78

13. In a test of hypothesis, the null hypothesis is that the population mean is equal to 50 and the alternative hypothesis is that the population mean is greater than 50.

The test is to be made at the 1% significance level. The critical value of Z is

(a) 2.07

(b) 2.33

(c) -2.58

(d) -1.96

14. The p-value is

(a) the largest significance level at which the null hypothesis can be rejected

(b) the largest significance level at which the alternative hypothesis can be

rejected

(c) the smallest significance level at which the null hypothesis can be rejected

(d) the smallest significance level at which the alternative hypothesis can be

rejected

15. For a one-tailed test, the p-value is given by

(a) the area under the curve between the mean and the observed area of the

sample statistic

(b) twice the area under the curve between the mean and the observed value of the sample statistic

(c) the area in the tail beyond the observed value of the sample statistic

(d) twice the area in the tail beyond the observed value of the sample statistic

16. For a two-tailed test, the p-value is given by

(a) the area under the curve between the mean and the observed area of the sample statistic

(b) twice the area under the curve between the mean and the observed value of the sample statistic

(c) the area in the tail beyond the observed value of the sample statistic

(d) twice the area in the tail beyond the observed value of the sample statistic

17) All first year students at the University of Irie are required to read a Foundation Course in Communications. For the purposes of Lectures, the first year intake of students are organized into two groups viz. Group A and Group B. Group B comprises students who have either an ‘A’ Level Pass or a Grade ‘A’ Pass at CXC General in English. All other students are assigned to Group A. However, both groups write the same final examination in the course.

The scores obtained in the course by two random samples of students, one from Group A and the other from Group B, are processed using MINITAB and some preliminary results are shown in Exhibit I below:

Exhibit I

Variable N Mean

Median

TrMean StDev SE Mean

Group A 135 51.44 51.00 51.40 09.37 *

Group B 108 51.80 53.00 52.05 ** 1.168

(a) Examine the results shown in Exhibit I and

(i) Fill in the gap marked * and ** [2 marks]

(ii) Estimate the proportion of students scoring less than 60 in Group A.

(iii)Estimate the highest mark made by the first 75% of students in Group A.

(b) Exhibit II below shows the results of further analysis carried out on the Group B scores.

Test of mu

Variable = 60 vs mu

N

not = 60

Exhibit II

Mean

StDev SE Mean T P

Group B 135 51.444 9.368 0.806 * **

95.0% CI = (49.850, 53.039)

(i) State the null and the alternative hypotheses for this test. [2 marks]

(ii) Calculate the missing value * and use tables to approximate the value of **.

(iii)If you were the statistician, what would you conclude and why? [2 marks]

18) It is claimed that the mean cost of a hotel room in Tobago is $125 US per day. A recent sample of 35 hotels showed that the mean cost of a hotel room is $138 US per day with a standard deviation of $26 US. Test at the 3% significant level if the current mean cost of a hotel room in Tobago is higher than $125US per day.

1. The mean of the sampling distribution of the sample mean is

(a) the mean of the means of all possible samples of the same size taken from the population

(b) the mean of the frequency distribution of the population

(c) the mean of the means of all frequency distributions

(d) the mean of one sample

2. To apply the central limit theorem to the sampling distribution of the sample mean, the sample is considered to be large if

(a) n is greater than 50

(b) n is 50 or larger

(c) n is larger than 40

(d) n is 30 or larger

3. The weights of all babies born at a hospital have a mean of 8.4 pounds and a standard deviation of 0.70 pounds. The mean of the sampling distribution of the mean weight of a sample of 49 babies born at this hospital is

(a) 1.2 pounds

(b) 8.4 pounds

(c) 0.10 pounds

(d) 7.0 pounds

4. The number of elements in a sample with a specific characteristic divided by the total number of elements in the sample is called

(a) the sample mean

(b) the sample proportion

(c) the sample distribution

(d) the sampling distribution

5. The value(s) assigned to a population parameter based on the value of a sample statistic is called

(a) the probability

(b) a probability distribution

(c) a sampling distribution

(d) an estimate

6. The Z value for a 95% confidence interval for the population mean is

(a) 2.07

(b) 1.96

(c) 2.17

(d) 1.65

7. A random sample of 100 customers who visited a department store spent an average of $77 at this store with a standard deviation of $19. The 97% confidence interval for the population mean is

(a) 74.56 to 79.44

(b) 70.18 to 83.82

(c) 73.89 to 80.11

(d) 72.88 to 81.12

8. A sample of 200 elements produced a sample proportion of 0.47. The applicable

Z score used in calculating an 98% confidence interval for the population

proportion is

(a) 2.323

(b) 0.765

(c) 1.388

(d) 2.862

9. In a test of hypothesis, the Type II error occurs when

a. a false null hypothesis is rejected

b. a true null hypothesis is not rejected

c. a false null hypothesis is not rejected

d. a true null hypothesis is rejected

10. A two-tailed test of hypothesis contains

(a) one rejection region and two nonrejection regions

(b) two rejection regions and one nonrejection region

(c) two rejection regions and two nonrejection regions

(d) one rejection region and one nonrejection regions

11. In a left-tailed test of hypothesis, the sign in the alternative hypothesis is

(a) not equal to ()

(b) greater than ()

(c) less than (<)

(d) less than or equal to ()

12. In a test of hypothesis, the null hypothesis is that the population mean is equal to 80 and the alternative hypothesis is that the population mean is less than 80. A

sample of 100 elements selected from this population produced a mean of 74 and

a standard deviation of 12. The value of the test statistic is

a. Z = 4.78

b. Z = 5.00

c. Z = -5.00

d. Z = -4.78

13. In a test of hypothesis, the null hypothesis is that the population mean is equal to 50 and the alternative hypothesis is that the population mean is greater than 50.

The test is to be made at the 1% significance level. The critical value of Z is

(a) 2.07

(b) 2.33

(c) -2.58

(d) -1.96

14. The p-value is

(a) the largest significance level at which the null hypothesis can be rejected

(b) the largest significance level at which the alternative hypothesis can be

rejected

(c) the smallest significance level at which the null hypothesis can be rejected

(d) the smallest significance level at which the alternative hypothesis can be

rejected

15. For a one-tailed test, the p-value is given by

(a) the area under the curve between the mean and the observed area of the

sample statistic

(b) twice the area under the curve between the mean and the observed value of the sample statistic

(c) the area in the tail beyond the observed value of the sample statistic

(d) twice the area in the tail beyond the observed value of the sample statistic

16. For a two-tailed test, the p-value is given by

(a) the area under the curve between the mean and the observed area of the sample statistic

(b) twice the area under the curve between the mean and the observed value of the sample statistic

(c) the area in the tail beyond the observed value of the sample statistic

(d) twice the area in the tail beyond the observed value of the sample statistic

17) All first year students at the University of Irie are required to read a Foundation Course in Communications. For the purposes of Lectures, the first year intake of students are organized into two groups viz. Group A and Group B. Group B comprises students who have either an ‘A’ Level Pass or a Grade ‘A’ Pass at CXC General in English. All other students are assigned to Group A. However, both groups write the same final examination in the course.

The scores obtained in the course by two random samples of students, one from Group A and the other from Group B, are processed using MINITAB and some preliminary results are shown in Exhibit I below:

Exhibit I

Variable N Mean

Median

TrMean StDev SE Mean

Group A 135 51.44 51.00 51.40 09.37 *

Group B 108 51.80 53.00 52.05 ** 1.168

(a) Examine the results shown in Exhibit I and

(i) Fill in the gap marked * and ** [2 marks]

(ii) Estimate the proportion of students scoring less than 60 in Group A.

(iii)Estimate the highest mark made by the first 75% of students in Group A.

(b) Exhibit II below shows the results of further analysis carried out on the Group B scores.

Test of mu

Variable = 60 vs mu

N

not = 60

Exhibit II

Mean

StDev SE Mean T P

Group B 135 51.444 9.368 0.806 * **

95.0% CI = (49.850, 53.039)

(i) State the null and the alternative hypotheses for this test. [2 marks]

(ii) Calculate the missing value * and use tables to approximate the value of **.

(iii)If you were the statistician, what would you conclude and why? [2 marks]

18) It is claimed that the mean cost of a hotel room in Tobago is $125 US per day. A recent sample of 35 hotels showed that the mean cost of a hotel room is $138 US per day with a standard deviation of $26 US. Test at the 3% significant level if the current mean cost of a hotel room in Tobago is higher than $125US per day.

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